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Answer to the Friday Puzzle….

Galileo pointed out that, ignoring air resistance, all bodies fall with the same acceleration. But what about objects sinking in water of different temperatures? Here is the puzzle: Suppose you drop a brick in each of two identical tanks. One of the tanks has water at 40 degrees F and the other has it at 30 degrees F. Which brick would sink faster?

If you have not tried to solve it, have a go now. For everyone else, the answer is after the break.

Well, the water at 30 degrees F would be frozen, so the brick would sink much faster in the other tank! Did you solve it?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

Of course the water could have been supercooled if it was very pure and the tank had no nucleation sites. Of course dropping the brick in would trigger the freezing action, but whether it would be freeze fast enough I don’t know. Anybody care to experiment? Not an easy test to do.

Just to confirm what Anders is saying with some real numbers. The latent heat of fusion of water is 334 kJ/kg. The specific heat is 4.2 kJ/kg/K. So to freeze water solid you need to shift about the same amount of heat to cool liquid water by about 75 K. As Anders says, if it was at -1C, very little of it could instantly freeze, and to get that kind of freezing it would have to be at around -20C. Notice that quite a lot of liquid remains after freezing but more than 1-2% of the water is freezing.

OK – I can see that you couldn’t get the instant freeze effect, but there might be enough ice forming at the boundary of the brick (and sticking to it) that it would decrease it’s average density.

Of course 40 F is very close to the maximum density of water, so the brick will be more “buoyant” in liquid water at 40F than at 30F, so if there is no “freezing effect”, the brick should fall faster in the super-cooled water (albeit I’ve no idea who much difference in viscosity there is).

Interesting how such an apparently simple problem has such physical complexities. But then water is a very odd chemical.

It is a darn interesting problem after all. Reminds me of the classic question of the many ways of measuring the height of a tower with a barometer …

I was at first thinking only in terms of viscosity, neglecting the density change. This is indeed justified as the viscosity changes much more strongly than density w.r.t. temperature (more than 10% per 10 F near 40 F). This means in the absence of freezing the fall is always faster in the warmer liquid. The caveat is that I do not know for sure if the viscosity trend with temperature extends into supercooled temperature range (getting higher with lower temperatures).

Now, 40 F is near 4 °C and we’re seeing an interesting density-effect as well due to the anomaly of water. The buoyancy is higher at 40 F, so the fall would be slower at 40 F if it were not counteracted by the lower viscosity.

Thirdly is the effect of accumulating ice when the brick is dropped into supercooled water. That would, as Steve pointed out, lower the brick’s average density, making it fall slower, or halting the fall entirely if the entire tank freezes up.

Both the effects of viscosity and partial or entire freezing go in the same direction as the canonical answer, slowing the fall at lower temperature. They are counter-acted by the density change, but to a much smaller degree (as it were).

It’s also been pointed out that the latent heat of fusion will release heat into the water as the brick is dropped, so this will warm up the liquid water and thus reduce the viscosity.

However, whilst the latent heat of fusion in normal circumstances is equivalent to heat liquid water by about 75K, I wonder if that figure applies to the freezing of super-cooled water? Even if it is anything like this value, I don’t think a high enough proportion of the ice could form to heat water significantly. Indeed I suspect we’d start hitting issues with the second law of thermodynamics if it did.

I suspect this would be a very difficult test to perform with a house brick as it is anything but a hydro-dynamically stable shape and sink times. especially over short distances, might be highly variable due to chaotic patterns of the attitude taken by the brick during sinking.

I feel any such test would require a body which would be more predictable in the way it sinks.

At first i figured that warm is lighter than cold, so the brick would be kind of lifted more from the sort of thicker cold water than by the lighter warm water. So the cold water would slow the brick the most. Later I remembered that 32 fahrenheit is 0 celsius.

Nice twist, I solved it quite quickly and it is a good example why we user ° Celcius instead of ° Fahrenheit, if it’s “-1” then it is obvious that it is frozen.

I had to use google to translate F to C and had first 40 ° F which was about 4 ° C where the density of water is highest, so before I translated 30 ° F to C I thought: “You sneaky bastard, if it is + 0.5 ° C or something then it’s density is lower and it would perhaps fall faster because of the anomaly of water.”

But then with -1 ° C it was obvious that you would not be able to get it to sink in ice.

You highlight an interesting point. Water is very unusual in that it increases in volume with freezing, a property it shares with very few other materials (bismuth is one). In consequence, unlike the vast majority of other materials, it can be melted by applying pressure. However, the pressures required are very high. For ice at 30F (just below -1C), a pressure of about 15MPa has to be applied. Normal atmospheric pressure is about 100KPa, so that’s roughly 150 times normal atmospheric pressure.

That then raises the point where such a test might be conducted.

Atmospheric pressure on Venus is about 92 times that of Earth, so that won’t do as a testing venue, and it’s far too hot anyway so refrigeration to 30F would be difficult. Fortunately, there’s a place where this could be tried far closer to home. At a depth of 1,500 metres in the oceans water pressure is approximately the require 15MPa. So, if out two tanks of water at 30F and 40F respectively were loaded into a diving bell and lowered one and a half kilometres down into the ocean, the experiment could be tried again. Of course the open diving bell would have to have it’s air pressure increased to prevent sea water incursion. Also, the pressure in the diving bell is far higher than any human can tolerate, so the whole experiment would have to be automated.

It would seem such an experiment is (surprisingly) technically feasible.

Water is at its densest at 4° Celsius. From that maximum, the density reduces both with colder and with warmer temperatures, so the brick should sink slightly faster in the colder tank but the difference is likely immeasurably small

there are at least four ways the bricks could sink at exactly* the same speed:

– highly saline water can have a freezing point below 30 deg F.
– supercooled water can remain liquid well below its freezing point.
– sufficiently low air pressure can also lower the freezing point.
– a sufficiently porous or spongy brick will float on the water that is 40 degrees.
– a brick heated to a sufficiently high temperature will melt the ice so quickly that it will sink through ice as quickly as through water.

Number 3 is wrong – because water expands when it freezes, you need higher pressure to lower the freezing point. In fact you need about 150 atmospheres to lower the melting point by 2 degrees Fahrenheit.

As for number 4 – I doubt it’s true. Anything that hot would undoubtedly vaporise both the liquid water and the ice. I doubt that with the explosive effect you’d have a brick left. That’s even assuming that you can heat the brick to a high enough temperature without it melting itself.

I’ve been scuba diving several times in -1C salt water and was thus convinced that the puzzle was about knowing the density quirks of water at 4C…
Had you stated that it was water from the tap I’d have thought your conclusion was the more likely solution.